What is moment of inertia and what does it measure?

Moment of inertia I is rotational mass: it measures how hard it is to change an object spin about a chosen axis. It depends on both the mass and how far that mass sits from the axis, so a 2 kg ring of radius 0.5 m has more inertia than a 2 kg solid disk of the same radius. The unit is kg per metre squared. The larger I is, the more torque you need to reach a given angular acceleration, which is why flywheels are built heavy at the rim.

How do I find the moment of inertia of a solid cylinder or disk?

For a solid cylinder or flat disk spinning about its central axis, I = half m r squared. A 2 kg disk of radius 0.5 m gives I = 0.5 times 2 times 0.25 = 0.25 kg per metre squared. Notice the height of the cylinder does not appear: about the long axis only the radius matters. This is also the formula a wheel or a pulley uses, which is why doubling the radius quadruples the inertia for the same mass.

Why is a solid sphere two fifths m r squared but a hollow shell two thirds?

A solid sphere has I = two fifths m r squared because much of its mass sits close to the centre, near the axis, where it contributes little. A thin spherical shell has I = two thirds m r squared because all of its mass sits out at the surface, far from the axis. For the same mass and radius the shell always wins: a 5 kg, 2 m solid sphere gives 8 kg per metre squared while the shell gives about 13.3. That gap is why a hollow ball rolls down a ramp slower than a solid one.

What is the difference between a rod about its centre and about its end?

A thin rod spun about an axis through its centre uses I = one twelfth m L squared, while the same rod spun about one end uses I = one third m L squared, exactly four times larger. The end axis is harder to turn because, on average, the rod mass sits farther from it. A 12 kg, 1 m rod gives 1 kg per metre squared at the centre and 4 at the end. The parallel axis theorem links the two: moving the axis from centre to end adds m times half L squared.

How does this calculator get rotational kinetic energy?

Once it has the moment of inertia I, the tool uses KE = half I omega squared, where omega is the angular speed in radians per second. It is the rotational twin of the linear half m v squared. Leave the angular speed blank and you just get I. Fill it in, say a 0.25 kg per metre squared disk at 4 rad/s, and you get KE = 0.5 times 0.25 times 16 = 2 joules. Remember omega is in radians per second, not rpm: divide rpm by 9.549 to convert.

What units does it use and can I use grams or centimetres?

The calculator works in SI base units: mass in kilograms, radius and length in metres, angular speed in radians per second, giving I in kg per metre squared and energy in joules. There is no built-in gram or centimetre switch, so convert first: divide grams by 1000 to get kilograms and divide centimetres by 100 to get metres. Because I scales with the square of the radius, a centimetre mix-up shifts the answer by a factor of ten thousand, so double-check the length unit before you read the result.

Solve rotational dynamics homework with the steps shown

A physics problem gives a 12 kg, 1 m rod and asks for the moment of inertia about its end, then the kinetic energy when it swings at 2 rad/s. Pick the rod-about-end shape, type the mass and length, and read I = 4 kg per metre squared with the substituted formula right below it. Add the angular speed and the rotational KE appears too. Because the worked formula is visible, you can copy the reasoning into your answer instead of just the final number.

Size a flywheel for an energy-storage estimate

You are sketching a flywheel and want to know how much energy it banks at a target speed. Model the rim as a solid disk, enter its mass and radius to get I, then fill in the running angular speed in rad/s. The tool returns the stored rotational kinetic energy, so you can trade radius against mass and see which change buys more energy per kilogram before you commit to a design.

Compare how shape changes the inertia of the same mass

Teaching why a hollow ball rolls slower than a solid one? Keep mass and radius fixed and flip between solid sphere and thin shell: the shell reads two thirds m r squared against the solid two fifths, a visible jump from 8 to about 13.3 kg per metre squared for a 5 kg, 2 m ball. The same trick shows a rod about its end carrying four times the inertia it has about its centre.

Check a CAD or simulation inertia value by hand

A CAD package or physics engine reports a moment of inertia for a part and you want a sanity check. Approximate the part as the nearest primitive — disk, tube, sphere or rod — plug in the mass and the governing radius or length, and compare. If your hand figure and the software disagree by an order of magnitude, you have probably caught a units mix-up such as millimetres entered as metres.

Moment of Inertia Calculator — Cylinder, Sphere, Rod

Pick a shape, enter mass and size, read I in kg·m² — cylinder, sphere, rod, point mass, plus rotational KE — browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Shape and axis

Mass m (kg)

Radius r (m)

Angular speed ω (rad/s)Optional. Fill it to also get the rotational kinetic energy.

Result

Moment of inertia I0.25 kg·m²

Formula

I = ½·m·r² = ½ × 2 × (0.5)²

What this tool does

Free moment of inertia calculator for physics homework, engineering design and rotational dynamics. Choose one of seven common rigid-body shapes and the tool applies the right formula: solid cylinder or disk uses I = half m r squared, a hollow cylinder uses half m times the sum of inner and outer radius squared, a solid sphere uses two fifths m r squared, a thin spherical shell uses two thirds m r squared, a thin rod about its centre uses one twelfth m L squared, the same rod about one end uses one third m L squared, and a point mass uses m r squared. Enter mass in kilograms and the radius or length in metres to get the moment of inertia I in kg per metre squared, with the substituted formula shown so you can check every step. Add an angular speed in radians per second and it also returns the rotational kinetic energy from KE = half I omega squared. One click copies the result and the formula, and the shareable URL reopens your exact shape and inputs. Everything runs in your browser, 100 percent client-side, nothing uploaded.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Moment of Inertia Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Solve rotational dynamics homework with the steps shown
A physics problem gives a 12 kg, 1 m rod and asks for the moment of inertia about its end, then the kinetic energy when it swings at 2 rad/s. Pick the rod-about-end shape, type the mass and length, and read I = 4 kg per metre squared with the substituted formula right below it. Add the angular speed and the rotational KE appears too. Because the worked formula is visible, you can copy the reasoning into your answer instead of just the final number.
Size a flywheel for an energy-storage estimate
You are sketching a flywheel and want to know how much energy it banks at a target speed. Model the rim as a solid disk, enter its mass and radius to get I, then fill in the running angular speed in rad/s. The tool returns the stored rotational kinetic energy, so you can trade radius against mass and see which change buys more energy per kilogram before you commit to a design.
Compare how shape changes the inertia of the same mass
Teaching why a hollow ball rolls slower than a solid one? Keep mass and radius fixed and flip between solid sphere and thin shell: the shell reads two thirds m r squared against the solid two fifths, a visible jump from 8 to about 13.3 kg per metre squared for a 5 kg, 2 m ball. The same trick shows a rod about its end carrying four times the inertia it has about its centre.
Check a CAD or simulation inertia value by hand
A CAD package or physics engine reports a moment of inertia for a part and you want a sanity check. Approximate the part as the nearest primitive — disk, tube, sphere or rod — plug in the mass and the governing radius or length, and compare. If your hand figure and the software disagree by an order of magnitude, you have probably caught a units mix-up such as millimetres entered as metres.

Common pitfalls

Mixing up which axis the formula is for. A rod about its centre is one twelfth m L squared, but about its end it is one third — four times bigger. Solid cylinder formulas here are for the long central axis, not for spinning end over end. Pick the shape entry that names your axis, or the number will be right for a problem you are not solving.
Entering length or radius in centimetres or millimetres. The tool works in metres, and moment of inertia scales with the square of that dimension, so a length keyed in centimetres lands the answer off by a factor of ten thousand. Convert to metres first, dividing centimetres by 100 and millimetres by 1000.
Using rpm or degrees per second for the angular speed. The kinetic energy formula needs omega in radians per second. Feeding it rpm inflates the energy by a factor of about 91 since rpm is roughly 0.105 rad/s each. Convert rpm to rad/s by dividing by 9.549 before you type it in.

Privacy

Every step — choosing the shape, applying the formula, computing I and the rotational kinetic energy — is plain JavaScript running in your browser tab. No mass, dimension or result ever leaves the page, and nothing you type is logged. The one caveat: the shareable URL encodes your shape and inputs in the query string, so a share link pasted into chat records those numbers in the recipient server access log. For anything you would rather keep private, use the copy button and paste the text instead of the URL.

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Moment of Inertia Calculator — Cylinder, Sphere, Rod

Result

Formula

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Moment of Inertia Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Solve rotational dynamics homework with the steps shown

Size a flywheel for an energy-storage estimate

Compare how shape changes the inertia of the same mass

Check a CAD or simulation inertia value by hand

Common pitfalls

Privacy

FAQ

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